I just realized that my comment actually answer completely the case of a Cartesian closed category with finite coproducts, and not just extensive categories. So I'm posting it as an answer.

The short version is that the sets you can get out of a Cartesian closed category as $\mathrm{Hom}(1,1 \amalg 1)$ are exactly the *Boolean algebras*, so the only finite cardinals you get are the $2^n$ and all infinite cardinals can be obtained.

Here is the longer version, with way too much detail:

Let $C$ be a Cartesian closed category with finite coproducts. As products are left adjoint, they commute to colimits, in particular coproducts, so that one has canonical isomorphisms:

$$ \left( A \times X \right) \amalg \left( B \times X \right) \overset{\sim}{\rightarrow} \left( A \amalg B \right) \times X $$

In particular:

$$ (1 \amalg 1) \times (1 \amalg 1) \simeq 1 \amalg 1 \amalg 1 \amalg 1 .$$

This allows to define all the "logical operations" $\vee$, $\wedge$, $\Rightarrow , \dots $ : $(1\amalg 1)^2 \rightarrow (1 \amalg 1)$, simply using the universal property of the coproduct above (one just have to specify the values of these functions on the four summands, which basically means giving their truth table) and using that more generally:

$$ (1 \amalg 1)^n= \coprod_{2^n} 1 $$

one can check that they satisfy all the expected relations (precisely, all the relations that can be checked on finite truth tables). This exactly makes $(1 \amalg 1)$ into a Boolean algebra object in $C$.

In particular:

**Proposition :**
*In a Cartesian closed category C, for any object $X$, the set $\mathrm{Hom}(X, 1 \amalg 1)$ has the structure of a Boolean algebra. In particular $\mathrm{Hom}(1, 1 \amalg 1)$ is a Boolean algebra.*

Now conversely, if I start with any Boolean algebra $B$, I can consider its Stone space $X$, which is a topological space, such that, among other things the Boolean algebra of clopen subsets of $X$ identifies with $B$.

In a topos of sheaf $\mathrm{Sh}(X)$, $1 \amalg 1$ is the sheaf of locally constant functions with values in $\{0,1\}$, i.e. of clopen subsets: sections of $1 \amalg 1$ over an open subset $U$ are exactly clopen subsets of $U$. So in particular in $\mathrm{Sh}(X)$,

$$ \mathrm{Hom}(1, 1 \amalg 1) \simeq B .$$ Hence:

**Proposition :**
*Any Boolean algebra $B$ appears as $\mathrm{Hom}_C(1 , 1 \amalg 1)$ for $C$ a Cartesian closed category, in fact for the Grothendieck topos $C =\mathrm{Sh}(\mathrm{Stone}(B))$.*

So in the end the sets you can get from this construction are *exactly* the Boolean algebra. All finite Boolean algebras are atomic, so of the form $\mathcal{P}(\{1,\dots,n \})$. In this case the Stone space is $\{1,\dots,n \} $ with the discrete topology and this is the example mentioned in the question. But because of the Lowenheim-Skolem theorem, there are Boolean algebras of any infinite cardinality.